Question: Imran and Aubrey were asked to find an explicit formula for the sequence $14,5,-4,-13,...$, where the first term should be $g(1)$. Imran said the formula is $g(n)=14-9(n-1)$. Aubrey said the formula is $g(n)=14-9n$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Imran (Choice B) B Only Aubrey (Choice C) C Both Imran and Aubrey (Choice D) D Neither Imran nor Aubrey
Solution: The general explicit formula for arithmetic sequences is ${a_1}+{d}(n-1)$, where ${a_1}$ is the first term and $ d$ is the common difference. The first term is ${14}$ and the common difference is ${-9}$. ${-9\,\curvearrowright}$ ${-9\,\curvearrowright}$ ${-9\,\curvearrowright}$ ${14},$ $5,$ $-4,$ $-13,...$ We get the following formula. $g(n)={14}{-9}(n-1)$ So Imran is definitely right. What about Aubrey? We can see that $g(n)=14-9n$ is not a correct formula, because the constant difference is added one extra time for each term. For instance, according to this formula, the value of the first term would be: $g(1)=14-9\cdot1=5$. However, according to our table of values, $g(1)=14$. So Aubrey is definitely wrong. Only Imran got a correct explicit formula.